~

Author: valbert4

codes/classical/analog/analog.yml

@@ -27,7 +27,7 @@ relations:
     - code_id: bosonic_classical_into_quantum
       detail: 'Any analog code can be embedded into a bosonic Hilbert space, and thus passed through a bosonic channel, by associating the reals with the configuration space of position states of bosonic modes.'
     - code_id: 2pt_homogeneous
-      detail: 'Euclidean space \(\mathbb{R}^n\) is a noncompact two-point homogeneous space and is, in fact, a noncompact three-point homogeneous space \cite[Sec. 6.6.1.1]{doi:10.1007/978-3-642-18245-7}.'
+      detail: 'Euclidean space \(\mathbb{R}^n\) is a noncompact two-point homogeneous space and is, in fact, a noncompact three-point homogeneous space \cite[Sec. 6.6.1.1]{doi:10.1007/978-3-642-18245-7}. One can also think of \(\mathbb{R}^n\) as a homogeneous space of the Euclidean group by the orthogonal group, \(E(n)/O(n)\) \cite[Ch. XI]{manual:{Vilenkin, N. I. (1978). Special functions and the theory of group representations (Vol. 22). American Mathematical Soc.}}.'
     # Not sure about C^n so its a cousin
 
 

codes/classical/analog/sphere_packing/lattice/dual/leech.yml

@@ -25,9 +25,9 @@ notes:
 relations:
   parents:
     - code_id: niemeier
-      detail: 'The Leech lattice is the Niemeier lattice with minimal norm 4 \cite{doi:10.1006/eujc.1999.0360}. Every Niemeier lattice is a sublattice of the Leech lattice \cite{arxiv:q-alg/9607013,doi:10.1006/eujc.1999.0360}.'
+      detail: 'The Leech lattice is the Niemeier lattice with minimal norm 4 \cite{doi:10.1006/eujc.1999.0360}. Every Niemeier lattice is a sublattice of the Leech lattice \cite{arxiv:q-alg/9607013,doi:10.1006/eujc.1999.0360}. In the holy construction for the Niemeier lattice \(A_2^{12}\), the combinations for which the sum of all coefficients is zero form a copy of the Leech lattice \cite[Ch. 24, pg. 510]{doi:10.1007/978-1-4757-6568-7}.'
     - code_id: construction_a4
-      detail: 'The Leech lattice can be constructed from pseudo Golay codes via \term{Construction \(A_4\)} \cite{doi:10.1016/S0012-365X(98)00358-6,arxiv:2309.02382}. The Leech lattice can be obtained by lifting the Golay code to \(\mathbb{Z}_4\) \cite{doi:10.1109/18.312154}, appending a parity check, and applying \term{Construction \(A_4\)} \cite{doi:10.1007/3-540-57843-9_20} (see also \cite{doi:10.1098/rspa.1982.0071,doi:10.1007/978-1-4757-6568-7}). Half of the lattice can be obtained in a different construction \cite[Exam. 10.7.3]{preset:EricZin}.'
+      detail: 'The Leech lattice can be constructed from pseudo Golay codes via \term{Construction \(A_4\)} \cite{doi:10.1016/S0012-365X(98)00358-6,arxiv:2309.02382}. The Leech lattice can be constructed from the extended quaternary Golay code via \term{Construction \(A_4\)} \cite[3rd Ed., pg. xxxiii]{doi:10.1007/978-1-4757-6568-7} (see also \cite{doi:10.1098/rspa.1982.0071,doi:10.1109/18.370138,arxiv:2309.02382}).'
     - code_id: univ_opt_analog
       detail: 'The Leech lattice is universally optimal \cite{arxiv:1902.05438}.'
   cousins:
@@ -43,14 +43,14 @@ relations:
       detail: 'The \(E_7\) lattice can be specified as a section of the Leech lattice \cite[Fig. 6.2]{doi:10.1007/978-1-4757-6568-7}.'
     - code_id: eeight
       detail: 'The \(E_8\) lattice can be specified as a section of the Leech lattice \cite[Fig. 6.2]{doi:10.1007/978-1-4757-6568-7}. The Leech lattice admits a Turyn-type construction from two copies of the \(E_8\) lattice in the same space \cite[Ch. 8, pg. 211]{doi:10.1007/978-1-4757-6568-7}.'
-    - code_id: extended_golay
-      detail: 'Two copies of the 24-dimensional packing obtained from the extended Golay code can be fitted together without overlap to form the Leech lattice \cite[Ch. 5, pg. 145]{doi:10.1007/978-1-4757-6568-7}. Half of the lattice can be obtained using Construction \(B^{\star}\) \cite[Exam. 10.7.3]{preset:EricZin}.'
     - code_id: combinatorial_design
       detail: 'The Leech lattice is completely determined by the Steiner system \(S(5,8,24)\) formed by the octads of the extended Golay code \cite[Ch. 12, pg. 335, Thm. 6]{doi:10.1007/978-1-4757-6568-7}.'
-    - code_id: golay
-      detail: 'The Leech lattice can be obtained by lifting the Golay code to \(\mathbb{Z}_4\) \cite{doi:10.1109/18.312154}, appending a parity check, and applying \term{Construction \(A_4\)} \cite{doi:10.1007/3-540-57843-9_20} (see also \cite{doi:10.1098/rspa.1982.0071}).'
+    - code_id: extended_golay
+      detail: 'Two copies of the 24-dimensional packing obtained from the extended Golay code can be fitted together without overlap to form the Leech lattice \cite[Ch. 5, pg. 145]{doi:10.1007/978-1-4757-6568-7}. Half of the lattice can be obtained using Construction \(B^{\star}\) \cite[Exam. 10.7.3]{preset:EricZin}.'
     - code_id: ternary_golay
       detail: 'A 12-dimensional complex version of the Leech lattice can be obtained from the ternary Golay code \cite{doi:10.1016/0021-8693(83)90074-1,doi:10.1098/rspa.1982.0071}\cite[pg. 200]{doi:10.1007/978-1-4757-6568-7}.'
+    - code_id: quaternary_golay
+      detail: 'The Leech lattice can be constructed from the extended quaternary Golay code via \term{Construction \(A_4\)} \cite[3rd Ed., pg. xxxiii]{doi:10.1007/978-1-4757-6568-7} (see also \cite{doi:10.1098/rspa.1982.0071,doi:10.1109/18.370138,arxiv:2309.02382}).'
     - code_id: sharp_config
       detail: 'Several spherical sharp configurations are derived from the Leech lattice \cite{arxiv:math/0607446}.'
     - code_id: octacode

codes/classical/analog/sphere_packing/lattice/dual/niemeier.yml

@@ -26,6 +26,8 @@ relations:
       detail: 'The nine inequivalent \([24,12]\) doubly even self-dual codes \cite{doi:10.1016/0097-3165(75)90042-4} yield certain Niemeier lattices via \term{Construction A} \cite{doi:10.1006/eujc.1999.0360}. Niemeier lattices can be constructed from ternary self-dual codes of length 24 \cite{doi:10.1016/0012-365X(93)E0088-L}.'
     - code_id: self_dual_over_zq
       detail: 'Extremal Type II self-dual codes of length 24 over \(\mathbb{Z}_6\) yield Niemeier lattices \cite{doi:10.1006/eujc.2002.0557}.'
+    - code_id: octacode
+      detail: 'The octacode is the glue code for the Niemeier lattice \(A_4^6\) \cite[3rd Ed., pg. liv]{doi:10.1007/978-1-4757-6568-7}.'
     - code_id: tetracode
       detail: 'The tetracode is the glue code for the Niemeier lattice \(E_6^4\) \cite[Ch. 16, pg. 408]{doi:10.1007/978-1-4757-6568-7}.'
     - code_id: hexacode

codes/classical/rings/over_zq/over_z4/linear_over_z4/self_dual/octacode.yml

@@ -44,9 +44,9 @@ relations:
       detail: 'The octacode is a cyclic code over \(\mathbb{Z}_4\) with generator polynomial \(x^3+3x^2+2x+3\) extended by a parity check \cite{doi:10.1007/3-540-30731-1}.'
   cousins:
     - code_id: hamming743
-      detail: 'The octacode can be obtained by Hensel-lifting the \([7,4,3]\) Hamming code to \(\mathbb{Z}_4\) \cite{doi:10.1109/18.312154}.'
+      detail: 'The heptacode can be obtained by Hensel-lifting the \([7,4,3]\) Hamming code to \(\mathbb{Z}_4\) \cite{arxiv:math/0311319,doi:10.1109/18.312154}.'
     - code_id: hamming844
-      detail: 'The mod-two reduction of the octacode is the \([8,4,4]\) extended Hamming code \cite{doi:10.1007/3-540-30731-1}.'
+      detail: 'The mod-two reduction of the octacode is the \([8,4,4]\) extended Hamming code \cite{doi:10.1007/3-540-30731-1}. The octacode can be obtained by Hensel-lifting the \([8,4,4]\) extended Hamming code to \(\mathbb{Z}_4\) \cite{arxiv:math/0311319}.'
     - code_id: eeight
       detail: 'The octacode yields the \(E_8\) Gosset lattice via \term{Construction \(A_4\)} \cite{doi:10.1007/3-540-57843-9_20,doi:10.1109/18.370138}\cite[Exam. 12.5.13]{doi:10.1017/CBO9780511807077}.'
     - code_id: projective

codes/classical/rings/over_zq/over_z4/linear_over_z4/self_dual/quaternary_golay.yml

@@ -25,11 +25,9 @@ relations:
       detail: 'The extended quaternary Golay code is an extremal Type II self-dual code over \(\mathbb{Z}_4\) by virtue of its parameters \cite{doi:10.1007/s10623-023-01293-7}.'
   cousins:
     - code_id: extended_golay
-      detail: 'Codewords of the extended quaternary Golay code with entries 0 and 2 are of the form \(2c\), where \(c\) is a codeword of the extended Golay code. Its mod-two reduction (mapping \(0,1,2,3\) to \(0,1,0,1\)) also yields the extended Golay code; see Ref. \cite{arxiv:2309.02382}.'
+      detail: 'Codewords of the extended quaternary Golay code with entries 0 and 2 are of the form \(2c\), where \(c\) is a codeword of the extended Golay code. Its mod-two reduction (mapping \(0,1,2,3\) to \(0,1,0,1\)) also yields the extended Golay code; see Ref. \cite{arxiv:2309.02382}. The quaternary Golay code can be constructed from the extended Golay code by Hensel lifting to \(\mathbb{Z}_4\) \cite[3rd Ed., pg. xxxiii]{doi:10.1007/978-1-4757-6568-7}\cite{arxiv:math/0311319}.'
     - code_id: combinatorial_design
       detail: 'Supports of codewords of any fixed symmetrized type of the extended quaternary Golay code form a 5-design \cite{doi:10.1016/S0012-365X(98)00035-1,doi:10.1002/(SICI)1520-6610(1998)6:3<225::AID-JCD4>3.0.CO;2-H,doi:10.1016/S0378-3758(99)00117-2}.'
-    - code_id: leech
-      detail: 'The Leech lattice can be constructed from the extended quaternary Golay code \cite{doi:10.1109/18.370138,arxiv:2309.02382}.'
 
 
 # Begin Entry Meta Information

codes/classical/rings/over_zq/q-ary_linear_over_zq.yml

@@ -15,7 +15,7 @@ description: |
   More technically, linear codes over \(\mathbb{Z}_q\) are submodules of \(\mathbb{Z}_q^n\).
 
   Linear codes can be defined using generator matrices \cite{doi:10.1007/s10623-008-9243-1,doi:10.1007/s10623-008-9220-8,doi:10.1090/S0002-9947-01-02905-1}.
-  There is a standard form of the generator matrix for even \(q\) and an upper-triangular permutation-equivalent standard form for \(q\) being a prime power \cite{doi:10.1007/BF01390768,doi:10.1109/18.761269,arxiv:math/0311319}.  
+  There is a standard form of the generator matrix for even \(q\) and an upper-triangular permutation-equivalent standard form for \(q\) being a prime power \cite{arxiv:math/0311319,doi:10.1109/18.761269,arxiv:math/0311319}.  
 
 notes:
   - 'See book \cite{doi:10.1007/978-3-319-59806-2} for an introduction.'